They allow you to make different creative decisions. Take Av (aperture mode), if you are creating a portrait, you will likely want a large aperture for a flattering shallow depth of field, so maybe you set it to f2.8 and let the camera figure out the best shutter speed.
However, if you want to create a landscape, you want most of the image in focus, so you ...

Correct. As a matter of fact, on most cameras Program Shift is the same too by your logic.
These modes, including Program, are designed to give you the same exposure which is why the results are the same since Aperture and Shutter-Speed are inversely related given a fixed ISO.
The difference is what you decide on. Only you can decide if you would like a ...

What you are going wrong is not giving the camera enough latitude. You fixed the aperture and ISO, so all the camera can do is set the shutter-speed and flash power. It must be not as low-light as you think because most often you would get an under-exposed image doing what you are doing.
The camera has a shutter-speed range it can use with the flash. The ...

Look at your exposure meter. Your gear is only capable of so much and in the Av/Tv/P modes it will attempt to use your settings (such as your specified shutter speed), and adjust the other settings to get the right exposure. If you adjust a setting too far then it won't be able to keep up. You should see your meter and/or shutter/aperture settings (whichever ...

First, I don't know, which advantages of Av/Tv you don't know about. :-)
For me, these modes are very useful, when I do not have time for full manual setting and I do not want to use full auto mode.
Sometimes for example I need the camera not to choose shutter speed less than 1/250 (when I use my telephoto zoom) but I am sure, the camera will choose ...

So why are there two different modes or what's the difference between them?
Because trying to control aperture by setting the shutter speed and trying to predict what aperture the camera will choose is impractical. When you're taking a portrait, for example, you might want to choose a specific aperture. You can adjust the shutter speed in Tv mode until ...

You are essentially correct in a simple case. It is really simply a convenience thing, though it does make a difference when dealing with more than 2 parameters. Don't think of it in terms of what the computer is doing, but what the person is doing.
If I want to have control of the depth of field for creative reasons, I will choose Av. If I happen to ...

I think what you're saying is that the camera will pick the same total exposure in either mode. For example: Suppose you're in aperture priority mode with f/8 selected. The camera picks a shutter speed of 1/250 second. If you then switch to shutter priority mode and select a shutter speed of 1/250 second, you find that the camera picks an aperture of f/8. ...

According to the Pentax K50 and K10D manuals, in P mode, if I select aperture, then program will choose a corresponding shutter speed, and if I specify shutter speed, then camera will select a corresponding aperture.
Not entirely true. After reading the manual, you can't set values other than ISO.
Av & Tv lets you set the aperture & shutter ...

It's a common mistake to use shutter priority because you want a slower shutter speed: you're thinking mainly about exposure time so having full control over the shutter speed seems the right way to go.
Unfortunately, the problem is that cameras have a vast range of shutter speeds (mine goes from 30 seconds to 1/8000th of a second) but a comparatively tiny ...

Additionally, if you're in Tv mode, when you half-press the button, if you see the Aperture number blinking, it means you are underexposing your picture. You can force the camera to take the picture, but the blinking was there to warn you.
In Manual mode, as was said by @tenmiles, the Exposure Meter will tell you whether you are properly exposed.
...

Short: In most cases "BULB" is a speed setting accessible only in MANUAL mode so you will have full control of aperture and ISO settings.
Longer:
"Bulb" mode is the ultimate manual mode.
Bulb is accessible in Manual mode and MAY be accessible in Shutter-speed priority mode.
It COULD have a setting of its own but is usually at the low end of the shutter ...

Most of the cameras that I know have separate Bulb mode and shutter priority, the Bulb can be in the manual mode and you can get it by increasing the shutter speed till you get bulb (Like in some Canon models). But it's not reasonable to make it attached with the shutter priority mode because the camera wouldn't know in advance how long you are going to open ...

The advantage of aperture or shutter priority modes over manual is the same as any automation: when it works right, it takes labor out of our hands and makes life easier, allowing us to concentrate on other things. And on modern cameras, the automatic metering is pretty good — it basically does the right thing most of the time. And, if you get to know your ...

You are getting pure white images because there is too much light coming into the camera.
For smooth water you need a shutter speed of at least half a second.
To reduce the amount of light coming into the camera in this time, you can do one or more of the following:
Reduce the ISO to the smallest value (e.g. ISO 100)
Make the aperture physically smaller ...

Well I was trying to shoot outside on a bright sunny day. I was
totally relying on the meter
The meter is for manual mode. Ignore it if you're in shutter or aperture priority.
It may blink Lo at you if it thinks it's too dark, or pop up the flash, but just give it a try and see what you get. You were probably fine to shoot
Sunny 16 Rule says on ...

In my experience (with a Canon, not Pentax, but it's probably similar), P mode is part-way between Auto and Av/Tv modes.
The difference between P and full auto is that you are given the option of making creative adjustments, if you want. The camera will use its auto algorithm (or something like it) to choose an aperture and shutter speed, and you can then ...

Your exposure gets basically determined by three things:
Aperture
ISO
Shutter-speed
These three properties together determine the amount of stops. There are tables as a general guideline for how many stops you need under certain circumstances. One example is given here.. According to this table, the most common bright daylight setting equals the number ...

TV stands for Time Value. This is basically to indicate that you can set the Shutter Speed manually and the aperture will set automatically depending on the other parameters you've set like the exposure and the ISO. This mode is very handy in case of bird photography, sports photography or anything where shutter speed needs to be constant.

You should use the Aperture priority mode, and keep the aperture the same for all shots, otherwise you could get some odd focus differences between shots. The only downside is if you have to go low on your shutter speed and pick up motion blur.
The easiest thing to do is use the camera settings that do multiple shots (usually -2,0, +2) in rapid succession, ...

It does not.
Once in BULB, no automatic exposure parameter applies. Bulb is found in Manual mode on the vast majority of cameras. On most others it is a selectable shutter-speed in Manual mode.
When you can select it in another mode, then the camera uses defaults. Actually, on the two cameras I know that accept Bulb in shutter-priority, the exposure time ...

The "advantage" of Av/Tv (and to some extent P) modes is that the camera takes care of correct standard exposure while you take care of the creative (artistic, effects) aspects of the shot.
The "advantage" of M mode is that you have total control (for good and bad) on the exposure as well as the creative aspects.

Aperture Priority (Av) and Shutter Priority (Tv) only produce the same result if you choose the corresponding value after changing modes.
Say you meter a scene in Av mode with f/5.6 selected and the camera selects 1/60 sec. If you switch to Tv mode, select 1/60 sec and meter the same scene, the camera will of course select f/5.6. But if you select 1/1000 ...

Have you checked if you are using a shutter time longer than expected? Or maybe Bulb mode?
In S mode the shutter time is shown as a highlighted number in the camera display, but if this number is shown with quotes then the time is in fact in seconds.
So a number like 30 means the shutter will be open for 1/30 of a second, but a number like 30" means the ...

I sent the question to pentax support and here is the answer:
S: Cust. wants to know what the difference is between P mode and Tv
and Av mode.
T: At P-mode the shutter speed and the aperture will be set
automatically. However, the cust. has control over the two values and
he can still adjust them. Av mode prioritizes the aperture, so cust.
...

The manual mentions that the values are restricted in P mode. It may be that it only allows one to be adjusted at a time or it may restrict how far you can adjust the setting (for example, you may not be able to set a value that would result in an over or under exposure.) The manual, unfortunately, does not appear to go in to more details about what the ...

Either way they are designed to give you a picture that is correctly exposed. But you may want to control the aperture or the shutter. As an example, if someone is running and you want there to be no blur in their movement, then using a faster shutter speed will freeze their movement. It will adjust the aperture accordingly, which will allow more light in to ...

As seen in the comments, it looks like your camera us unaware of the flash, at least in some operating modes. Research in your user's manuals, there may be an answer. If not, Maybe your camera/flashgun combo is not able to use TTL (trhough the lens) flash regulation.
Another possibility is that some of the hotshoe contacts is not working (i.e. dirty) and ...